MathObj.Polynomial.Core

This module implements polynomial functions on plain lists. We use such functions in order to implement methods of other datatypes.

The module organization differs from that of ResidueClass: Here the Polynomial module exports the type that fits to the NumericPrelude type classes, whereas in ResidueClass the sub-modules export various flavors of them.

Documentation

horner :: C a => a -> [a] -> a

Horner's scheme for evaluating a polynomial in a ring.

hornerCoeffVector :: C a v => a -> [v] -> v

Horner's scheme for evaluating a polynomial in a module.

hornerArgVector :: (C a v, C v) => v -> [a] -> v

normalize :: C a => [a] -> [a]

It's also helpful to put a polynomial in canonical form. normalize strips leading coefficients that are zero.

shift :: C a => [a] -> [a]

Multiply by the variable, used internally.

unShift :: [a] -> [a]

equal :: (Eq a, C a) => [a] -> [a] -> Bool

add :: C a => [a] -> [a] -> [a]

sub :: C a => [a] -> [a] -> [a]

negate :: C a => [a] -> [a]

scale :: C a => a -> [a] -> [a]

collinear :: (Eq a, C a) => [a] -> [a] -> Bool

tensorProduct :: C a => [a] -> [a] -> [[a]]

\(QC.NonEmpty xs) (QC.NonEmpty ys) -> PolyCore.tensorProduct xs ys == List.transpose (PolyCore.tensorProduct ys (intPoly xs))

tensorProductAlt :: C a => [a] -> [a] -> [[a]]

mul :: C a => [a] -> [a] -> [a]

mul is fast if the second argument is a short polynomial, ** relies on that fact.

mulShear :: C a => [a] -> [a] -> [a]

\xs ys  ->  PolyCore.equal (intPoly $ PolyCore.mul xs ys) (PolyCore.mulShear xs ys)

mulShearTranspose :: C a => [a] -> [a] -> [a]

divMod :: (C a, C a) => [a] -> [a] -> ([a], [a])

\x y -> case (PolyCore.normalize x, PolyCore.normalize y) of (nx, ny) -> not (null (ratioPoly ny)) ==> mapSnd PolyCore.normalize (PolyCore.divMod nx ny) == mapPair (PolyCore.normalize, PolyCore.normalize) (PolyCore.divMod x y)

\x y -> not (isZero (ratioPoly y)) ==> let z = fst $ PolyCore.divMod (Poly.coeffs x) y in  PolyCore.normalize z == z

\x y -> case PolyCore.normalize $ ratioPoly y of ny -> not (null ny) ==> List.length (snd $ PolyCore.divMod x y) < List.length ny

divModRev :: (C a, C a) => [a] -> [a] -> ([a], [a])

The modulus will always have one element less than the divisor. This means that the modulus will be denormalized in some cases, e.g. mod [2,1,1] [1,1,1] == [1,0] instead of [1].

stdUnit :: (C a, C a) => [a] -> a

progression :: C a => [a]

differentiate :: C a => [a] -> [a]

integrate :: C a => a -> [a] -> [a]

integrateInt :: (C a, C a) => a -> [a] -> [a]

Integrates if it is possible to represent the integrated polynomial in the given ring. Otherwise undefined coefficients occur.

mulLinearFactor :: C a => a -> [a] -> [a]

alternate :: C a => [a] -> [a]

dilate :: C a => a -> [a] -> [a]

shrink :: C a => a -> [a] -> [a]